Definition 2.1 (see [[1]}, Definition 5.1] and [[2]}, Definition 2.4 and 2.6])
Let \(\mathcal {X}\)  be a Deligne-Mumford stack over \(S\)  with a moduli scheme \(\pi :\mathcal {X}\rightarrow X\) . A locally free sheaf \(\mathcal {E}\)  is said to be a generator for the quasi-coherent sheaf \(\mathcal {F}\)  if the left adjoint of the identity \(\pi _{*}(\mathcal {F}\otimes \mathcal {E}^\vee )\xrightarrow{}\pi _{*}(\mathcal {F}\otimes \mathcal {E}^\vee )\) , i.e.,
\(\theta _{\mathcal {E}}(\mathcal {F}): G_{\mathcal {E}}\circ F_{\mathcal {E}} (\mathcal {F}) =\pi ^{*}\pi _{*}\mathcal {H}om_{\mathcal {O}_{\mathcal {X}}}(\mathcal {E},\mathcal {F})\otimes \mathcal {E}\rightarrow \mathcal {F}\) 
